The statement i.e. `(n+3)^(2) gt 2^(n+3)` is true.

Options

A. For all n
B. For all `n ge 3`
C. For all `n ge 2`
D. No, `n in N`

Answer

To solve the inequality \((n+3)^2 > 2^{(n+3)}\), we will analyze the expression step by step.

### Step 1: Understand the inequality

We need to determine for which values of \(n\) the inequality \((n+3)^2 > 2^{(n+3)}\) holds true.

### Step 2: Test small values of \(n\)

Let's start by testing small integer values for \(n\).

#### Test \(n = 1\):
\[ LHS: (1 + 3)^2 = 4^2 = 16 \]
\[ RHS: 2^{(1 + 3)} = 2^4 = 16 \]
Here, \(16 \not> 16\). So, the inequality does not hold for \(n = 1\).

#### Test \(n = 2\):
\[ LHS: (2 + 3)^2 = 5^2 = 25 \]
\[ RHS: 2^{(2 + 3)} = 2^5 = 32 \]
Here, \(25 \not> 32\). So, the inequality does not hold for \(n = 2\).

#### Test \(n = 3\):
\[ LHS: (3 + 3)^2 = 6^2 = 36 \]
\[ RHS: 2^{(3 + 3)} = 2^6 = 64 \]
Here, \(36 \not> 64\). So, the inequality does not hold for \(n = 3\).

#### Test \(n = 4\):
\[ LHS: (4 + 3)^2 = 7^2 = 49 \]
\[ RHS: 2^{(4 + 3)} = 2^7 = 128 \]
Here, \(49 \not> 128\). So, the inequality does not hold for \(n = 4\).

### Step 3: Test larger values of \(n\)

#### Test \(n = 5\):
\[ LHS: (5 + 3)^2 = 8^2 = 64 \]
\[ RHS: 2^{(5 + 3)} = 2^8 = 256 \]
Here, \(64 \not> 256\). So, the inequality does not hold for \(n = 5\).

#### Test \(n = 6\):
\[ LHS: (6 + 3)^2 = 9^2 = 81 \]
\[ RHS: 2^{(6 + 3)} = 2^9 = 512 \]
Here, \(81 \not> 512\). So, the inequality does not hold for \(n = 6\).

### Step 4: Analyze the trend

From the tests above, we see that as \(n\) increases, the left-hand side \((n+3)^2\) grows quadratically, while the right-hand side \(2^{(n+3)}\) grows exponentially.

Given that the exponential function grows faster than the quadratic function, we can conclude that:

\[ (n+3)^2 < 2^{(n+3)} \quad \text{for all } n \geq 1 \]

### Conclusion

The inequality \((n+3)^2 > 2^{(n+3)}\) does not hold for any integer values of \(n\). Therefore, the correct option is that there are no values of \(n\) for which the inequality is true.